A Noahide teacher told a friend, who's a convert, that he's not really Jewish because one who converts has to bring a korbon thus he's not Jewish... he asked me regarding this topic but I personally never heard of such thing. Is there a korbon needed for conversion?

Similarly, for [all] future generations, when a gentile desires to enter into the covenant, take shelter under the wings of the Divine presence, and accept the yoke of the Torah, he must undergo circumcision, immersion, and the offering of a sacrifice. A woman [who converts] must undergo immersion and bring a sacrifice, as [Numbers 15:15] states: "As it is for you, so shall it be for the convert." Just as you [entered the covenant] with circumcision, immersion, and the offering of a sacrifice; so, too, for future generations, a convert must undergo circumcision, immersion, and must bring a sacrifice
What is the sacrifice that a convert [is required to bring]? A burnt offering of an animal or two turtle-doves or two fledging doves. Both of [the doves] must be brought as burnt offerings. In the present age, when there are no sacrifices, [a convert] must undergo circumcision and immersion. When the Temple is rebuilt, he must bring a sacrifice.

There are four individuals who are referred to as "requiring atonement": a zavah, a woman after childbirth, a zav, and a person afflicted by tzara'at. Why are they referred to as "requiring atonement"? Because even after each one of them has become pure from the condition that caused his impurity, he has immersed in a mikveh, and the day of the immersion has passed, the person's status is still lacking. His attainment of purity is not complete to the extent that he may partake of sacrificial foods until he brings the sacrifice required of him. Before he brings this sacrifice, he is forbidden to partake of sacrificial food as explained in Hilchot Pesulei HaMukdashim.
When a convert becomes circumcised, but has not brought his sacrifice, although he is forbidden to partake of sacrificial food until he brings his sacrifice, he is not considered as one requiring in atonement. It is merely that failing to bring his sacrifice prevents him from being a complete convert and being from the full-fledged members of the Jewish people. Accordingly, he may not partake of sacrificial foods, because he has not become a full-fledged member of the Jewish people. Once he brings his sacrifice and becomes a full-fledged member of the Jewish people, he may partake of sacrificial foods in the evening.

There is a Machloket if Ger is can eat Korban. Rabbi Eliezer Ben Yaacov indeed thinks that he cannot before he makes his Korban (two Olot Haof) (see Tosfot Yom Tov), but the first Tana of the Mishna did not aggree , and did not count him in the list. The first Tana ruled that Convert can eat Korbanot after circumcision and Mikve (some mefarshim discusse this opinion of Bartenura and Rambam in comment of Mishnayot, Beer Sheva, Aruch Laner, Rambam in Mishne Torah), but he cannot marry a Jewish woman until he makes his Korban Ola.

The Melechet Shelomo summarizes Gemara (disagree with Bartenura who says that Rabanan permit him to eat Korbanot before his own Korban), Gilayon Shita Mekubetset and Rabenu Gershom Maor Hagola. He says that Rabanan think that to marry a Jewish woman, convert needs a Korban. Rabenu Gershom Maor Hagola says that Rabbi Yehuda Hanasi in Berayta rules against Rabbi Eliezer Ben Yaakov (it's important to say because it's an exception to the rule, generally we pasken as REBY because his Mishna is "kav venaki"). Ramban too in his comment writes that the Halacha is not as R.E.B.Y.

Yachin:

וקיי"ל דבזה"ז שאין מקדש, משמל וטבל מותר בקהל [י"ד רס"ח ס"ב]

The Tiferet Israel says that nowadays, that we have no Mikdash, convert can marry a Jewish woman despite the lack of Korban. The source is in Gemara 9a.

If converts need a Korban to marry a Jewish (man/woman), how can they marry with Jews nowadays? Rabbi Acha Bar Yaakov learns from the verse that when we have no Mikdash, a stranger can convert, because this Halacha to convert is permanent through generations. This Gemara is reported by Bach in Y.D 268.

In summary: You can say to your friend that indeed Teacher's words has source in Gemara, both Rabbi Eliezer Ben Yaaqov and rabbanan say that convert must make a korban to join Jewish community at the time of Bet hamikdash. However, the Halachical conclusion of Gemara is that is not relevant nowadays, a convert is 100% truely Jew despite the lack of Korban. When Bet hamikdash will come back, he will give two Olot Haof.